Behold!

The above picture is my favourite proof of Pythagoras' theorem.
Filling in the details is left as an exercise to the reader.

Disclaimer:
I have learned quite a bit about this and other proofs of the Pythagoras
theorem since last time I edited this page.
I now know that much of what you read below is wrong or misguided.
Until I can find the time to improve the page,
you should read this with a skeptical eye.
(Always good advice anyhow.)
It's not all wrong, of course.
But to give just one example of the wrongness,
the Chou pei suan ching and Zhoubi suanjing are one and the same:
They are just transliterations of the Chinese phrase

周

髀

算

經

in, respectively, the
Wade–Giles
and the pinyin systems
of transcription
(the pinyin version should be zhōubì suànjīng, really).
(Disclaimer added 2005-07-21; it may still take weeks for me to get around
to a major overhaul of this page.)

Is this the oldest proof?

This proof is sometimes referred to as the Chinese square proof,
or just the Chinese proof.
The righthand picture above appears in the
Chou pei suan ching (ca. 1100 B.C.E.),
for the special (3,4,5) pythagorean triple.
See also Development of Mathematics in Ancient China.

The controversy over who had the first proof will probably last forever.
Part of the reason is that the notion of what is considered proof
changes with time. Hence, rather than obsessing over who was first, let us
instead throw away our prejudices and marvel at the ingenuity and analytic
abilities of our distant ancestors.

Friberg, a leading authority on Babylonian mathematics,
presented convincing evidence that the old Babylonians were aware of the
Pythagoras theorem around 1800 B.C.E.
In the clay tablets from the time one finds many examples of calculations
of a geometric nature which depend heavily on Pythagoras.
Moreover, they had a geometric proof of the algebraic identity
(a+b)2=a2+b2+2ab
which is essentially obtained by contemplating the left picture above.
(Of course, they did not write it algebraically as I did here,
but thought of the squares as real geometric objects, and also
2ab as two a×b rectangles.)
They were also very adept at generalizing from known results and
computing areas by moving bits around to arrive at better known areas,
so there is little doubt that they could have found the above proof.
Friberg is convinced that they did, though there is no firm evidence
of this. It should be recognized that the Babylonians had no concept of
axiomatization and abstract proof as we know them from Euklid.
Instead, they were absolute masters at all kinds of practical calculations.

Friberg also presented evidence of Babylonian influence on
Greek mathematics (indirectly, via the Egyptians).
Among other things, Pappus has a simplified version of Euklid's proof
of the Pythagoras theorem which seems influenced by Babylonian methods -
although he, like Euklid, uses shear transforms to distort rectangles
to parallellograms of equal area and back, which is very un-Babylonian.

Disclaimer: The above is just my interpretation of what Friberg
told us in the lecture.
I may well have misunderstood or misrepresented some points,
so don't blame him if I wrote something blatantly wrong.
(Later, Friberg told me that I have indeed not represented him accurately
in all respects, but he did not elaborate.
So take the above with quite a large grain of salt.)